Theorem undif5 4475

Description: An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024.)

Assertion

Ref	Expression
undif5	⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴)

Proof of Theorem undif5

Step	Hyp	Ref	Expression
1		difun2 4473	. 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
2		disjdif2 4472	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)
3	1, 2	eqtrid 2783	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∖ cdif 3938   ∪ cun 3939   ∩ cin 3940  ∅c0 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-nul 4316

This theorem is referenced by:  fressupp  31776  sucdifsn2  36893